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Topology and its Applications
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Tychonoff expansions with prescribed resolvability properties

Authors: Comfort, W.W.; Hu, Wanjun;

Tychonoff expansions with prescribed resolvability properties

Abstract

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) and to Comfort and García-Ferreira (2001): (1) Is every $ω$-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of ${\mathcal{KID}}$ expansion, the authors show that {\it every} suitably restricted Tychonoff topological space $(X,\sT)$ admits a larger Tychonoff topology (that is, an "expansion") witnessing such failure. Specifically the authors show in ZFC that if $(X,\sT)$ is a maximally resolvable Tychonoff space with $S(X,\sT)\leqΔ(X,\sT)=κ$, then $(X,\sT)$ has Tychonoff expansions $\sU=\sU_i$ ($1\leq i\leq5$), with $Δ(X,\sU_i)=Δ(X,\sT)$ and $S(X,\sU_i)\leqΔ(X,\sU_i)$, such that $(X,\sU_i)$ is: ($i=1$) $ω$-resolvable but not maximally resolvable; ($i=2$) [if $κ'$ is regular, with $S(X,\sT)\leqκ'\leqκ$] $τ$-resolvable for all $τ<κ'$, but not $κ'$-resolvable; ($i=3$) maximally resolvable, but not extraresolvable; ($i=4$) extraresolvable, but not maximally resolvable; ($i=5$) maximally resolvable and extraresolvable, but not strongly extraresolvable.

25 pages, 0 figures

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Keywords

Extremal set theory, 05A18, 03E05, 54A10, 03E35, 54A25, 05D05, resolvable space, \(\omega \)-resolvable space, independent family, FOS: Mathematics, Cardinality properties (cardinal functions and inequalities, discrete subsets), ω-Resolvable space, Mathematics - General Topology, Several topologies on one set (change of topology, comparison of topologies, lattices of topologies), strongly extraresolvable space, General Topology (math.GN), Strongly extraresolvable space, Maximally resolvable space, Other combinatorial set theory, Resolvable space, Partitions of sets, extraresolvable space, Extraresolvable space, Independent family, maximally resolvable space, Consistency and independence results, Geometry and Topology, Souslin number

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
3
Average
Average
Average
Green
hybrid